Because if string theory didn't include gravitons, we would make a new theory that did include them. Providing a quantum theory of gravity is one of the major goals of string theory. Why would somebody invent and study a theory that didn't even include the phenomenon they were interested in?

We assume it because it is the whole point of the theory! It's like asking why newtonian gravity assumes that an attractive force exists between massive bodies; it assumes it because quantifying it is the whole purpose of the theory.

And we have the "right" to assume it because the only important requirement on a theory is that it correctly predicts experimental results. You may form a theory based on magical pixy dust if you can show that it is able to predict observations, and do so more accurately than other theories, or at least do as well as other theories while having some other benefit such as simplicity.

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In this fourth Lecture of his string course , Lenny Susskind explains in a slightly technical and very accessible manner, why there is a spin 2 excitation for closed strings which can be interpreted a graviton.

To explain this, he writes down all the lowest possible excitations of a closed string.There can be left and right moving waves of different frequencies. Taking into account the so called level matching condition, which says that the energy of the right moving waves must match the energy of the left moving waves, all possibilities but the ones with spin 2 and spin 0 are excluded. Physically, the level matching corresponds to the fact that the translation of the wave function along the closed string is a symmetry transformation. It means that when treating a closed string as a sequence of mass points, there is no special point on the string (they are all equivalent).

As I understand it so far, the interpretation of the spin 2 excitation as the graviton can be motivated by the following considerations: When discribing gravity as a QFT in a curved spacetime, the metric tensor takes the role of the gauge field. Because it has two Lorentz indices (the vector potential in QED has only one) this gauge field (the graviton) must be spin 2. In addition, in an interacting string theory, closed strings can not be avoided. If the coupling constant does not vanish, the ends of open strings can always come together and join such that the strings get closed. The revers process can happen too (with the same coupling constant), such the all (open and closed) strings can absorb and emmit closed strings or gravitons respectively. This corresponds to the fact that everything gravitates.

... so if something wags its tail like a dog, barks like a dog, etc then it probably IS a dog :-)

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